Solved – How to interpret a negative linear regression coefficient for a logged outcome variable

interpretationlinear modelregression coefficients

I have a linear regression model where the dependent variable is logged and an independent variable is linear. The slope coefficient for a key independent variable is negative: $-.0564$. Not sure how to interpret.

Do I use the absolute value then turn it into a negative like this:
$(\exp(0.0564)-1) \cdot 100 = 5.80$

or

Do I plug in the negative coefficient like this:
$(\exp(-0.0564)-1) \cdot 100 = -5.48$

In other words, do I use the absolute figure and then turn that into a negative or do I plug in the negative coefficient? How would I phrase my findings in terms of a one-unit increase in X is associated with a __ percent decrease in Y? As you can see, these two formulas produce 2 different answers.

Best Answer

You should not take the absolute value of the coefficient--although this would let you know the effect of a 1-unit decrease in X. Think of it this way:

Using the original negative coefficient, this equation shows the percentage change in Y for a 1-unit increase in X:

(exp[−0.0564*1]−1)⋅100=−5.48

Your "absolute value" equation actually shows the percentage change in Y for a 1-unit decrease in X:

(exp[-0.0564*-1]−1)⋅100=5.80

You can use a percentage change calculator to see how both of these percentages map onto a 1-unit change in X. Imagine that a 1-unit change in X were associated with a 58-unit change in linear Y:

  • Our linear version of Y going from 1,000 to 1,058 is a 5.8% increase.
  • Our linear version of Y going from 1,058 to 1,000 is a 5.482% decrease.
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